Seamless grunge texture overlay across two materials
A way to add a grunge texture to both the board and the letters with no discontinuity between the 2 would be to join your 2 objects (make sure that you've applied the Scale and that the normals are recalculate), unwrap  with Smart UV Project mode, paint your grunge texture on both with the Stencil brush option and use the result as a mask:

But the quickest way to is probably to use the grunge image in an Image Texture that you switch from Flat to Box, which allows the texture to be projected on the 3 dimensions, and use the Texture Coordinate &quot;Object&quot; output socket, with an empty as Object in order to control the location and scale of the texture so that you can precisely put the grunge:


Then use this node chain as a mask in both the letter and the board materials. Here for the board I use it to simply separate 2 nuances of red:

Former answer, in case your letters were printed and part of the same object:
You don't say enough about your current material, but let's suppose that is very basic, like this one (a simple mix between 2 colors with a b&amp;w mask for the text):

then you can plug a dirt mask into 2 ColorRamp in order to get the dirty texture:

If your material is more complicated, for example because it plays with glossiness etc, it will make the node setup more complicated as well: